Paper · Dec 21, 2016

The Kähler-Ricci flow and optimal degenerations

Ruadhaí Dervan, Gábor Székelyhidi

math.DGarXiv:1612.07299v5
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Abstract

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a Kähler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.

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Submitted
Dec 21, 2016
Last revised
Jul 9, 2018 (v5)
Subjects
math.DG, math.AG

Comments: 14 pages, published version

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryStability & uniquenessKähler–Einstein & K-stability

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)5Hultgren 2017 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)8Liu 2019 The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations (Ryosuke Takahashi, 2019)4Takahashi 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)2Guo 2020 Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties (Jiyuan Han and Chi Li, 2020)3Han 2020 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)1Wang 2020 Lower bound of modified K-energy on a Fano manifold with degeneration for Kähler-Ricci solitons (Liang Zhang, 2021)7Zhang 2021 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)6Chen 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano manifolds”, “Chen Sun”, “Tian Zhu”

  2. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano manifolds”

  3. Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

    Jiyuan Han and Chi Li

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Kähler geometry · Shared terms “optimal degeneration”, “Sun Wang”, “Chen Sun”

  4. The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations

    Ryosuke Takahashi

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “optimal degeneration”, “Fano manifolds”

  5. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano manifolds”, “Tian”

  6. On Kähler-Einstein Currents

    Yifan Chen, Shih-Kai Chiu, Max Hallgren et al.

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Gábor Székelyhidi

  7. Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons

    Liang Zhang

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “Fano manifolds”, “degeneration”

  8. Stability of the conical Kähler-Ricci flows on Fano manifolds

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “Fano manifolds”, “flow converges”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.